The sum of two numbers is $103$, and their difference is $19$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 103}$ ${x-y = 19}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 122 $ $ x = \dfrac{122}{2} $ ${x = 61}$ Now that you know ${x = 61}$ , plug it back into $ {x+y = 103}$ to find $y$ ${(61)}{ + y = 103}$ ${y = 42}$ You can also plug ${x = 61}$ into $ {x-y = 19}$ and get the same answer for $y$ ${(61)}{ - y = 19}$ ${y = 42}$ Therefore, the larger number is $61$, and the smaller number is $42$.